The community organizer in the White House is using the current border crisis as a spending opportunity where enforcement of America’s threatened perimeter hardly appears at all: most items on the President’s want list are about providing services to illegal aliens rather than defending the nation.

Federal officials last week paid $584 to buy four versions of Monopoly, five packs of playing cards, a dozen beanbag tosses and other games for illegal immigrant children — a small fraction of what the Obama administration now says it will take to clothe, feed, transport, house and, yes, entertain the children surging across the border.

Government contractors stand to see a good payday after President Obama submitted a request Tuesday for an additional $3.7 billion this year to cover the costs of handling the illegal immigrant families and, most costly, the unaccompanied children from Central America who are entering the U.S. in record numbers.

“There are significant resources that are required to deal with the situation,” White House press secretary Josh Earnest said. “What we would like are some additional resources to make sure that we’re enforcing that law efficiently, both in terms of processing them through the immigration system, but also in terms of repatriating them if it’s found that they don’t have a legal basis for remaining in the country.”

Mr. Obama sent a letter to Congress detailing a request for 2014 alone: $879 million in additional funding to detain and deport illegal immigrant families, $116 million to transport unaccompanied children, and $522 million for more law enforcement both on the border and in Central America.

Mr. Obama also is asking for an additional $1.8 billion for the Health and Human Services Department to be able to house the children while they are awaiting immigration court hearings; $300 million for the State Department try to create better economic and security conditions in Honduras, Guatemala and El Salvador; and $64 million for the Justice Department to hire more immigration judges.

The request puts an initial dollar figure on the border crisis but does little to settle the heated dispute over the way Mr. Obama is handling it.

“President Obama created this disaster at our southern border, and now he is asking to use billions of taxpayer dollars without accountability or a plan in place to actually stop the border crisis,” said House Judiciary Committee Chairman Bob Goodlatte. “Most of the money requested in the president’s supplemental seems geared toward processing Central Americans than stopping the surge itself.”

Immigrant rights advocates, however, said they feared too much money was being used to beef up enforcement — particularly in trying to detain illegal immigrants to make sure they show up for their court dates and eventual deportation. They said the money would be better spent on lawyers to represent the children in their immigration cases.

But the White House, in a stark acknowledgment, said its enforcement has fallen short in recent years.

“The bottom line here is, the number of children removed is not large enough,” a White House official said in briefing reporters about the request for more money.

The White House said it will ask Congress for changes in law that would allow Mr. Obama to quickly deport children from Central America. Right now, that power is reserved only for children from the contiguous countries of Canada and Mexico, but the White House said that makes no sense.

The extra detention space and more teams of judges are designed to speed up the time it takes to process and deport illegal immigrants. Immigration authorities have testified to Congress that some of the children’s cases can take five years, which removes much of the deterrent effect of the administration’s pledge that illegal border crossers will be caught and returned to their home countries.

The government still can’t say how many of the more than 50,000 children from Central America who have been caught along the border this year have been deported.

The largest chunk of the requested money would go to organizations to house the children for the Health and Human Services Department.

Those companies have collected $1.4 billion over the past 14 years, according to federal contracting records, but could stand to make more than that this year alone.

The companies, however, said they had been ordered by the government not to speak to the press about their work, and an HHS spokesman would not say why.

Kenneth J. Wolfe, deputy director of the Office of Public Affairs, said the details of the work are included in the contracts with the companies, but those contracts had to be obtained under open-records requests. Those requests can take months to fulfill.

In the meantime, federal spending databases give a glimpse of past work and the skyrocketing costs already incurred this year.

HHS spending through the first nine months of the fiscal year totaled $366.7 million, marking a huge increase over just a few years ago, when the government spent $136.2 million for all of 2011.

From 2000 through 2014, one company, Southwest Key Programs Inc. has collected more than $328 million in taxpayer contracts, while Baptist Child & Family Services has taken another $232.9 million.

One major challenge has been transferring the 10,000 or so children caught at the border each month to other locations to be processed, then sending them to the HHS-sponsored facilities to be held. The White House spending request includes $116 million for transportation costs.

Before the children are turned over to HHS, they are held by Homeland Security authorities — as are the Central American families also surging across the border.

The Homeland Security Department has recorded tens of thousands of dollars in expenses for clothing, shoes, toiletries, translation costs, office equipment and “housekeeping,” according to spending records.

The children’s games were purchased for use at the detention facility being set up at the Federal Law Enforcement Training Center in Artesia, New Mexico. The $584 tab, racked up at Staples, covered four Monopoly games, six ring tosses, 12 beanbag tosses, 12 tic-tac-toe games, five packs of playing cards and eight checkerboards.

The spending records also show the difficulties of finding places to house some of the children.

One company signed an agreement to run a facility in Lawrenceville, Virginia, providing “shelter and mass care support” for unaccompanied minors. That plan was canceled after local residents protested.